
For \(\lbrace 0,1\rbrace \) -valued sequences, we show that Cover's UP can be simplified to a universal two-horse race scheme, which can be implemented with constant complexity per round to construct time-uniform confidence intervals.
This observation is simple, but has not been realized in the literature.

For \([0,1]\) -valued sequences, we discuss how to exactly compute the wealth of any mixture of wealth processes of constant bettors (such as Cover's UP) with \(O(t)\)  complexity at round \(t\)  and show that it leads to time-uniform confidence intervals.
To circumvent the cumbersome \(O(t)\)  per-round complexity of the UP, we propose a new UP-like method based on a new lower bound of the wealth of Cover's UP with constant complexity per round.
blackOur algorithm is based on Lemma REF , a higher-order generalization of the proof technique of [1]}, which may be of independent interest. Experiments show that the proposed algorithm gives tightest confidence sequences for all practical purposes.

